Wasserstein distances and divergences of order $p$ by quantum channels

TL;DR

This paper generalizes quantum optimal transport to non-quadratic cases using quantum channels, introduces p-Wasserstein distances and divergences, and explores their geometric properties, including a triangle inequality for certain divergences.

Contribution

It presents a novel non-quadratic framework for quantum optimal transport, extending previous quadratic models with new geometric insights.

Findings

Introduces p-Wasserstein distances and divergences for quantum channels.

Proves triangle inequality for quadratic Wasserstein divergences with a pure state.

Establishes fundamental geometric properties of the new distances.

Abstract

We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.